An exotic version of this type of space, homeomorphic but not diffeomorphic to the standard type, was constructed by Milnor in dimension 7. All of these spaces can be realized
An exotic version of this type of space, homeomorphic but not diffeomorphic to the standard type, was constructed by Milnor in dimension 7. All of these spaces can be realized as iterated suspensions of the space of two points. The Hopf fibration exhibits a covering of a 2-dimensional one by a 3-dimensional one whose fiber is a 1-dimensional one of these. The Poincaré-Hopf degree theorem says that maps from a manifold to one of these are classified up to homotopy by their degree, and a standard example of such a map is the Gauss map. Smale demonstrated that a two-dimensional one can be turned inside out without forming creases in this type of eversion. In 3 dimensions the Poincare conjecture states that every simply-connected closed manifold is homeomorphic to one of these. For 10 points, name this space, which may be defined as the subset of Euclidean space given by all points of unit distance from the origin.
An exotic version of this type of space, homeomorphic but not diffeomorphic to the standard type, was constructed by Milnor in dimension 7.
spheres